CE Z3 A) Find all the values that make the Z[x]/(x² +x + c) quotient ring an object. B) Show that the polynomial f(x) = x4 – 2x³ + 3x2 + 1 given in Z[x] is also irreducible in

Q: 1) let f(x) be a polynomial with integer coefficients satisfying f(2021) = 2020. assume that f(x)…

A: Given that f(2021) = 2020. Also given that f can be factored into five polynomials with integer…

Q: that 1+2i is a zero of the polynomial f with real coefficients. Which of the following statements is…

A: Given query is to find the incorrect statement.

Q: Let 0 be a root of the polynomial x³ + 3x + 7. Prove that 0 is prime in Q(0).

A: We will use the basic knowledge of polynomials and rings to answer this question properly, in…

Q: In Z3[x], show that the distinct polynomials x4 + x and x2 + xdeterminethe same function from Z3 to…

A: We know that ℤ3 is a field i.e., a commutative ring with unity in which every non-zero element has a…

Q: . Factor x12 -1 as a product of irreducible polynomials over Z.

A: Given: x12-1. We have to factor x12-1 as a product of irreducible polynomials over Z.

Q: Consider the following cubic polynomial function: p(x) = 2x32x29x - 5 a) How many zeros does this…

A: a)  3b)  Nc)  -14Explanation:Thank you 😊

Q: A real number x ∈ R is called algebraic if there exist integersa0, a1, a2, . . . , an ∈ Z, not all…

A: An= the set of zereos of polynomials with integer coefficents of degree atmost n.

Q: What is a polynomial in x?

A: The x happening in a polynomial is generally referred to as a variable or an indeterminate. When the…

Q: Show that every polynomial of degree at least 1 takes every complex value a E C.

A: We have to prove that every polynomial of degree atleast 1 takes every complex value a in C .

Q: Whether the polynomial x² + 2x + 2 over Z3, Z5, and Z7 is irreducible or not and state all zeros.

A:

Q: C) Find a polynomial of degree 2 that has two irrational zeros

A:

Q: ) Consider the polynomial f(x) = x4+1 (a) Explain why f has no real roots, and why this means f…

A: f(x)= x4+1 let x2=y=>x4=y2 So, f(x)= y2+1 General quadratic polynomial ay2+ by +c on comparing…

Q: (4) Find a polynomial that has 1+i and 1- i as zeros.

A: We have to find polynomial

Q: consider the polynomial P(x)= x3-x2-4x + 4 a) by factor thereom, show that (x+2) is a factor b) is…

A:

Q: 5. (a) State the Fermat's theorem. (b) Use Fermat's theorem to find the remainder of 83 when it is…

A: 5. (a) To State: Fermat's Theorem   (b) To Use: Fermat's theorem to find the remainder of 8103 when…

Q: Write x3 + 3x² + 3x + 4 in Zs[x] as a product of irreducible polynomials.

A:

Q: If 2-i is a complex zero of P(x)= x* – 3x – x² +13x-10, find ALL zeros of the polynomial.

A:

Q: Show that the given values for a and b are lower and upper bounds for the real zeros of the…

A: Solution:-

Q: Let P be a polynomial of odd degree, prove that the range of P is all real numbers.

A: Let p(x)=xn+an-1xn-1+………………+a1x+a0 is polynomial of degree n. For any real number A>0, such that…